You can write a book review and share your experiences. Other readers will always be interested in your opinion of the books you've read. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them.

Lecture Notes in Mathematics
Edited by A Oo!d and 8 Eckmann
416
Michael Taylor
Pseudo Differential Operators
Spnnger-Verlag Berlin. Heidelberg' New York 1974
Dr. Michael E. Taylor University of Michigan Ann Arbor, MI 48104/USA
Library of Congress Cataloging in Publication Data
Taylor, Michael Eugene, 1946- Pseudo differential operators.
(Lecture notes in mathematics ; 416) Bibliography: p. Includes index. 1. Differential equations, Partial. 2. Pseudo- differential operators. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 416. QA3.L28 no. 416 [QA374] 510'.8s [515'.724] 74-23846
AMS Subject Classifications (1970): 35-02,35S05
ISBN 3-540-06961-5 Springer-Verlag Berlin' Heidelberg' New York ISBN 0-387-06961-5 Springer-Verlag New York' Heidelberg' Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photo- copying machine or similar means, and storage in data banks. Under 9 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. @ by Springer-Verlag Berlin' Heidelberg 1974. Printed in Germany.
Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
Introduction Chapter I. l. 2. 3. Chapter II. l. 2. 3. 4. 5. 6.
TABLE OF CONTENTS
1
Singular Integral Operators on the circle
4
The algebra of singular integral operators The oblique derivative problem * C algebras and singular integral operators
6
10
14
Pseudo Differential Operators 19
The Fourier integral representation 19
The pseudo local property 22
Asymptotic expansions of a symbol 24
Adjoints and products 31
Coordinate changes, operators on a manifold 33
Continuity on H S
37
7. Families of pseudo differential operators 41 8. Garding's inequality 44
Chapter III. Elliptic and Hypoelliptic Operators 45 1. Elliptic operators 4S
2. Hypoelliptic operators with constant strength 48
3. References to further work 57
Chapter IV. The Initial Value Problem. Hyperbolic Operators 58
1. Reduction to a first order system 59
2. Symmetric hyperbolic systems 62 3. Strictly hyperbolic equations 66
4. Finite propagation speed
finite domain of dependence 72
5. 6. 7. Chapter V. 1. 2. 3. 4. 5. Chapter VI. l. 2. 3. 4. Chapter VII. l. 2. 3.
IV
The vibrating membrane problem 76
Parabolic evolution equations 79
References to further work 82
Elliptic Boundary Value Problems; Petrowsky Parabolic Operators 84
A priori estimates and regularity theorems 91
Closed range and Fredholm properties 98
Regular boundary value problems 107
A subelliptic estimate; the oblique deriva- tive problem 115
References to further work 119
Propagation of Singularities; Wave Front Sets 120
The wave front set of a distribution 120
Propagation of singularities; the Hamilton flow 125
Local existence 131
Systems
an exponential decay result 135 The Sharp Garding Inequality 139
A multiple symbol 140 Friedrichs symmetrizgtion l¥
The sharp Gerding inequality 147 Bibliography 150
LECTURES ON PSEUDO DIFFERENTIAL OPERATORS
INTRODUCTION: These notes are based on the lectures I gave in partial differential equations at the University of Michigan during the winter semester of 1972, with some extensions. References to further work have been added at the end of Chapters III, IV, and V, and a few exercises have been thrown in, in addition to those thrown out in class. The students to whom these lectures were addressed were assumed to have knowledge of elementary functional analysis, the Fourier transform, distribution theory, and Sobolev spaces, and such tools are used without comment. We refer the reader especially to Yosida [85] for the background ma
erial. TDe last section of Chapter 1 also relies on the basic results of C* algebra theory, and the reader who doesn't like func- tional analysis might have to skip this section on first reading. Beyond that, we have tried to make these notes self-contained. This is not to say that these notes constitute a self contained introduction to the subject of partial differential equations, and the beginning student would have to see the material in several of the books we have mentioned in the references, especially, [13], [25], [31], and [1], [52], [54], [55], [59], in order to get a good idea of what the subject is about. What we do here is develop one tool, the calculus of pseudo differential operators, and apply it to several of the main problems of partial differential equations.
-2-
We begin in Chapter I by describing the earliest sort of singular integrals on the circle investigated by poinca
,
Hilbert, and others, and an application to the oblique deri-
vative problem on the disc. In the second Chapter we intro-
duce the modern calculus of pseudo differential operators, II developed by Kohn and Nirenberg, Lax, Hormander, Kumano-Go,
and others, and in Chapter III we apply these results to
obtain interior regularity results for elliptic and hypo-
elliptic operators. The next two chapters are devoted to
the main topics of classical PDE, the initial value problem
for hyperbolic and parabolic equations, and boundary value
problems for elliptic equations. We give a unified treat- ment of these topics, and Gardingi; inequality plays a crucial role here in passing from formal properties of symbols to the
energy inequalities and other a priori inequalities
eeded for
various results on existence and regularity. In Chapter VI we cover some recent work of Hormander
on wave front sets and the prop
gation of singularities of
solutions to partial differential equations. Applications are
given to local existence of solutions to PDE's and to an
exponential decay result. The proof of the main result on propagation of singularities requires the sharp 8arding
inequali ty which, follCMing Kumano-Go, we prove in the last
chapter of these notes.
One important topic we bave not included is Uniqueness in the Cauchy problem. W6 recommend th
.t the reader consult [Cfl].
-3-
It is a pleasure to thank Eric Bedford, whose class- room notes greatly aided the preparation of these notes, and Professor Jeff Rauch for some interesting conversations, especially relating to hyperbolic equations.
HAPTER I. SINGULAR INTEGRAL OPERATORS ON THE CIRCLE
The basic singular integral operator with which we will
If
be concerned here can be described as follows.
L 2 (Sl)
, write
u e
Pu :,[ n=O
u ( , and define n
n=-oo
00
a n
in e
In order to interpret the operator P, which is clearly a continuous orthogonal projection on L2(Sl) , as a singular
integral operator, consider the Cauchy integral
T".l(z)
21Ti
1
We can rewrite this as 1T i6 1 f u(d ei-z
d
d + u(8)
If u e C l(Sl), we can pass to the limit as r ->- 1 and obtain
Um
i6 T u(re ) =
I 1T -1T
u( + U ( 8 )
1 2iT
-s- lim 1 f u( + u (8) 2iT 1 i( 8-1- E:+O Sl"I (8) -e E: lim 1 I u(1;) - u(8) d1;: + u(8) E:+O 21Ti 1;-e 8 81" .,.. E: (8) 1 PV f u(1;) 1 21T i i 8 d1; + 2" u (8) 1;-e
Sl \Vil"'T'0 T (
) = ($ - [. (,oJ + [ ). f. Since it is easy to verify that
i8 lim Tu(re ) rtl
Pu for
] 1* U e C' (S
) ,
it follows that Pu 1 Hu + 1. u where 2" 2 Hu(e i8 ) 1 PV f u(1;) d1; ni 1;-e i 8 S'
The singular integral operator H is called the Hilbert
transform. The formula we have just de
ived shows that H extends to a continuous line,r 0perator onL 2 (sl).
Exercise 1. Find the Fourier series representation of H. Prove that H is a unitary operator on L2(
) and H 2 = I .
II< (using the :r:'esidue t28orA,I1. the re3der shOll1.d Ch9(
k that. . ike l6..r '/ /"\ If uk ;;; e t then lim '1"J..h, (l'e' ) = u- K lJ. k -- O. 0 if k ..... 0.)
-6-
91. The algebra of singular integral operators. Definition: The algebra Cl of singular integral operators on S 1 is the norm closed algebra of operators on L 2 (sl)
generated by:
(1) P (2) multiplication by a e C(Sl) (3) (; , the set of compact operators
Actually (3) is redundant, but we shall not prove this fact,
nor make use of it.
Theorem 1: If A, B em, then [A,B] = AB - BA e r Proof: It suffices to show that ap - PA e [ if
a e C (S.J..) .
Suppose that
im a = e ,
f
00 r:
in a e n
Then
apf
eim
t n=O
a ein n
00 1:
a n-m
n=-oo ein
j and
n=m
Paf = p(t
00
The [a,P]f
in
n=O
a n-m
in e .
Hence [a,P]
in an operator
n=O
with finite dimensional range, and therefore is compact. Since trigonometric polynomials are dense in cts l ), the result holds for all a e C (S.J..), because [.. is norm closed. This theorem, which says that
is commutative, modulo C
is important in that it enables us to give a nice conditon that
-7-
an operator in Ol be Fredholm. For the moment, consider an operator T e 01. of the form
T = aP + b(l-P) + K K e [ I
In section 3 we shall show that every T e tJL is of this form,
but we won't need this, since all singular integral operators
ofieencounters are automatically constructed in this form. For
such aT, we tentatively define the symbol aT of T, as
a function on
sJ x
?l
by L
aT (,l) = a(,-l) = b(f (8)
f(-8), the map + U lal 2 PU_
is
continuous
in the uniform operator topology, sin
e
U" lal 2 PU f(8) .: la(8-
1 21T
f 21T ulal2 PU_ d= Ilall
P e [, which
forces a - 0 .
o
Similary we obtain b = 0 .
Theorem 2: If T = aP + b(l-P) + K 1 , and W
aP + S(l-P) + K 2 '
then aT Ow a TW
-8-
Proof: This is immediate from the computation
TW (aP + b(l-P) + K 1 ) (aP + S (l-P) + K 2 ) 2 2 aaP + bS(l-P) + K3
aaP + bS(l-P) + K3
Recall that a linear operator T e L(L 2 ) is called
Fredholm if
(1) R(T) is closed
(2) dim ker T < 00
(3) dim coker T < 00
The reader should also recall the following important result from the Riesz theory of compact operators (see [64], Chap. VII) Proposition: T e
(L2) is Fredholm if and only if there
exists
U e t:.. (L 2 ) , called a Fredholm inverse of T, such that
TU = I + K l , and UT = I + K 2 ' where Kl ' K 2 are compact. The following Fredholm property of singular integral opera-
tors is now immediate.
Theorem 3: Let T = aP+ b(l-P) + K. Then T is Fredholm
if
aT
is nowhere vanishing.
Proof.
Let U =
P +
(l-P).
Then
a TV
OUT - 1 ,
so U is a Fredholm inverse of T.
In section 3 we shall show that this conditon on aT is also necessary for T to be Fredholm on L 2 (S')
The problem of how to define the symbol of a singular
integral operator on a multidimensional space took quite scrne time in being solved. Mikhlin defined a symbol in 1936. This
-9-
symbol was elucidated by Calderon and Zygmund in their important
works in the early 1950's. It was Lax who suggested a Fourier
series representation to treat multidimensional singular integrals,
and the Fourier integral representation used by Kohn and Niren-
berg is the one we shall use in the next chapter.
Exercise:
et Tf (x) where a € C Oo (Sl X Sl).
1 = 1T i Snow
J-:V ( a(x..ZL ) S I y-x that
fCy) dy.
Tf = bHf + F:! where b(x) = 8.lx,x) and K:H s -7 H S is co.:Jpact, for all s.
-10-
92. The oblique derivative problem.
Here we discuss one application of the algebra of sin-
gular integrals developed above. For further applications we
refer the reader to Mikhlin [58] and Muskhelishvili [60]
The problem we consider is the oblique derivative problem for
functions harmonic on the disc: 1 find given g e C (8""' ) , u harmonic on B = {z e !I:: I z I n
Then
n = -00
00 pIf(rei r e n n - 00 1 f 1 - 2 r f(6) d8 2iT 2 l+r - 2r cos (-8) -1T
The reader can verify as an exercise thet
If does t
trick. Exercise 2. Prove that if s > -
and f e ff (S )
-11-
"",-1:. w or 2 then PIf e H (B). too s0riously.) In view of the fact that restriction to
1 T-- H 2
for
1 T > '2
I (S -)
with S > 0 .
where we define T
u = PI h if Th
inverting T
\. ')U first .i.'celdill G , C:Oh't take tllis exercise
T S' maps H (B)
onto
, we have the following commutative diagram
PI oj. S -+ l- IHs · IB) H S (SJ,)
SCPI. Hence we can solve (1) by setting
g
Hence solving (1) is equivalent to
More generally, we are interested in when problem (1) is
Fredholm, in the sense that it can be solved provided g
satisfies a certain finite number of linear conditions, and the
sional.
set of u satisfying
1il = 0 , Su = 0 should be finite dimen-
This is equivalent to asking when is T Fredholm, which is
right up our alley, since we will now write T as a pseudo
differential operator. We have
u = PIf =
a: u ISl
t
Inl in r e
a n
n - 00
00 a f Lin ein a;p a n n - 00
-12- 00 and u Is. L Inl ein a n n = -t>
Now let us define an operator A by A{
) 00 in in I: (1 + I nl) a e a e n n _00 n = - 00
Exercise 3. Prove that A:HS(Sl) + HS-l(S:) isomorphically,
for each real S
It follows that T d + b a. C = a(A-l) + b d + C a- ar+ ar dr d A- l + -1 (a + b V (c-a) A ) A
Since A is an isomorphism of HS+l(S') onto H S (S. ) it follows that T:HS+l(Sl) S ' is Fredholm if and only if + H (S
-) (a + b d A- l + -1 H S + H S is Fredholm. S = ar (c-a) A ) :
Exercise 4. A- l : H S + H S is compact, for each real S.
Now we compute that
d A- l a;p
r an ein
i(2P-l)
([
a in an ein O. The sa
e condition suffices, as can be seen from
the following proposition. Proposition: If a e COO (S'), then AT a A- T -a is compact L 2 (S') I for any real
on
T .
Here
AT
r:
a ein
We omit the
proof of this proposition here, as it is a very simple consequence
of more general results proved in the next chapter. The reader
might want to try to find a direct proof.
*
If a and b are smooth real valued functions, then
d a- dr
d + b
is a vector field on
S} , and the Frelhol
condition becomes the condition that this vector field not vanish
at any point in
1 S.l. .
In higher dimensions this oblique
derivative problem becomes more subtle, and things can go
wrong if the vector field is tangent to the boundary at some points. We will take a look at this in Chapter v.
*
Alternatively, the Fredholm theory we
rked out on L 2 (8 1 ) can easily be pushed to H S (Sl) .
-14-
93. C* algebras and singular integral operators.
The basic task of this section is to prove the necessity
of the Fredholm condition given in section 1. We do this
using an approach to pseudo differential operators due to
Cordes; see [12], [29].
For singular intergral operators on a half line and half
space, see [10], [11], and for an algebra of operators
related to hypoelliptic operators see [76].
With all due apologies to the reader, we begin with the
following list of algebras of operators we'll need. S = norm closed algebra generated by I and P . 0.= algebra generated by P and C(S') . 07, norm closure of (Jl 0
I
t (i Cil,
'V
Ol. algebra generated by 0(0 and r.
01.
AI norm closure of 01
Now tDeorem 1 of section 1 implies that the C* algebra
l/I is commutative. Hence it is isometrically isomorphic
to C(M), the space of continuous complex valued functions on
its maximal ideal space. Since 01 1 is the Ba!'.ach algebra generated by 6-' and C (8"') , if we consider the injection jl == 0/-+ 01} and j2: C(S..) -+ Oi,' t t ' then adjoints lead to maps of maximal ideal spaces j
x j2:M-+S.X.
-15-
This map must be 1-1 since J? and C (S: ) generate 01 1 . It is also onto. In fact, rotational symmetry of Sl ShOW3 th
t
(,E) e M
implies
(' ,E) e M
for any ' e S", E= +1, and m 2 shows that (,l) e M
(,-l) eM.
flipping Sl about an axis in I Thus 07 l /I
C (S'" x R 2 ). Proposition: Ol. III :::::::OV C; in a natural manner Proof: Consider the natural maps OJ 1
V{ --10/ c: .
ClearJ..:v the kArnel of this Cortlposite map is precisely 0(, n c:. = I , so we have an injective map K;
I I -7 01/ c:. Now observe that the range of K .:::> Cl/ t , which is dense in (Jf / Csince ifL is
dense in C(. But a*-homomorphism of C* algebras must have
closed range (sec Dixmier [16] .) So K is the desired natural
isomorphism.
image 1 x
) A e 01 under Now the in C(S" of an elemen:t the composite map 01 -+- Ol It
Cll l / I
C(Sl x 2'l;z ) 1. fJ c.' 1 1(''-:1 the symbol of A and is denoted by °A The reader should verify that °a(,E) = a( ,1) - 1, 0p ( ,-1) = 0 ,
so this agrees with the definition of 0A given in section 1 . (an immediate cOlcllary is that every A eqcan be written in the form used in section 1.)
Reillo.rl
:
Since an isomorphism of C* algebras preserves the norm,
we have the following consequence. If A e
, then
inf II A+K II Ke z:-
sup I ° A (, E) I ( , E ) eS 'x ?l 2
-16-
With one further result from C* algebra theory, we'll be ready to finish off the question of when is A e OlFredholm.
Proposition: If X is a C*algebra with identity I and Y
is a C* subalgebra, containing I, then A e Y is invertible
in Y if and only if it is invertible in X
The reader who is familiar with the first chapter of
Dixmier [16] should be able to supply the proof of this.
Now for our lli
iL theorem.
Theorem:
A e Otis Fredholm if and only if
°A
is nowhere
vanishing.
Proof: We know that A is Fredholm if and only if it is invertible in.L (L 2 ) / C. In view of the above quoted result, this is equivalent to A being invertible in
/
, and since A .... a A is an isomorphism of O( / t with C (Sl x ?l2)' the result follows.
Exercise 5. If T:X .... Y is a Fredholm operator between two Banach spaces, we define the index of T to be ind T dim ker T - dim (Y/T(X) )
dim ker T
dim ker T*
F(X), the set of Fredholm operators on X , is an open subset of ,
(X), with the norm t
pology, and the index is constant on connected
components of F(X).
(See [64].)
, If T = aP + b (l-P), where a, b e C (8""") are non-vanishing
complex valued functions, calculate ind T in terms of the
winding numbers of a and b (about 0), using the above facts and explicitly calculating ind T when a(z) = ZD , b(z) = zm
( I z 1=1), m, n e ?l
-17-
Exercise 6.
For
1 s > - 2 ' consider the map
T:H S + 2 (B) -+- H S (B) G)H S +
(S 1 ) defined by
u
,
d d (ll u, a ar- u + b a;p u
+ cu
Is 1. )
If the Fredholm
condition given in section 2 is satisfied, T is Fredholm.
In particular, if
a
d ar
+ b
d a;p
is a real, nonvanishing
vector field, find ind T in terms of the winding nu
ber of
this vector field.
L 2 (S ')
Exercise 7: The range of P which we will call H 2 -
i
: a closed linear subspace of
If

= P
I H 2
is an operator on H 2 called a Toeplitz operator. (a) Show that the linear map C(Sl) -+- £(!!2) given by cp -+- T
, induces a linear * homomorphism of the algebra C(SL) into 1. ([,2) / r:, where r in the algebra of compact operators on H 2 .
(b) Show that T is corr.pact if and only if cp = 0 Hint" T compac
T Icpl2 compact. Now imitate the proof of lemma 1 in section l. (c) Deduce that if;! is the C*algebra of operators on H 2
generated by 11f:p: cp e C (S')} and C the map
-isomorphism of cIs l} -+- J / C .
-.. T cp
induces a
(d)
Prove that IITcp II Hint: Clearly IITcpll.:::..
sup, xeS"
! cp (x)
-1 for cp e C (8-- )
II

is Fredholm if and only if , e C(S') . Calculate the index
is nowhere vanishing, for
of
T .
(f) L. Coburn [9] has shown that if cp e c(Sl) (more generally, if e Loo(S')) is not identically zero, then either T
or
* T
must be injective.
Using this fact and the result
of exercise (e), deduce that T is invertible if and only if is nowhere vanishing and the curve traced out by
has winding number zero
about the origin. More generally,
describe the spectrum of T
HAPTER II. PSEUDO DIFF
NT
AL 2PE
TORS
l. The Fourier integral representation.
Recall that the Fourier inversion formula is
f(x) (21T)-n J ei f(l;;)dl;; , f e C;(m. n )
and if we differentiate this formula we get, with
1 D. ==-r J
a ax:- J
Daf(x) == (21T)-n Jr I;;a ei
" f (I;;) dl;;
Hence if
p(x,D)
L
a (x) D a a
is a differential operator,
(1)
p(x,D)f(x)
I a I < k n J i ,. (21T)- p(x,l;;) e ' f(l;;) dl;;
We shall use the Fourier integral representation (1) to define
pseudo differential operators, taking the function P(x,l;;) to belong to a general class of symbols , which we now define.
Definition: Let n be an Opfl
subet of JR n . Then
m,p,SeJR,
m a
f' 0 < 1 , sp,o (1:)
is the set of p e c"'(n x JRn)
with the property that for any compact Ken , any multi-indices
a and 6, there exists a constant cK,a '
such that
(2)
ID 6 x
D
P(x,l;;) I
c (1 + I I;; I ) m- p I a I + 0 I 6 I K,a,B
for all xeK, all I;; e
-20-
If P e Sm (r2) we say p is a symbol on r2 of order p , t5 I!! , ar.d type (p, t5) . If p satisfies (2) only for I
I > R , - we say p e Sm for large
. p,t5
Exercise 1. If p(x,D1 is a differential operator of order
m, show that
m p(x,
) e Sl,O
If p(x,
)
... n e C (r. >. (r. 'O))is homogeneous of degree m p e Sl,O for large
m
in
, prove that
Exercise 2.
m (a) peS (r2)
P (B) p,t5
ilslDS Da sm-plal + t5ISI x
pe p ,t5 (r2).
m] m+ml (r2) If q e S (r2) , then p(x,
) q(x,
) e 5 p ,t5 p,t5 If P e sO (r2) and cP is smooth in a nbd of the closure of the p,t5 cP (p (x,
) ) 0 set of values assumed by p, then e S t5(r2) . p,
Theorem:
If
p e Sm (r2) then p(x,D) defined by formula (1) p,t5
is a continuous map of C
(r2) into C'" (r2) map can be extended to a continuous map
If t5 < 1 , then the
p(>:,D):
' (r!)
.jt (r2)
Proof: If P e S
,t5(r2), u e C
(r2), then the integral
-1 p(x,D)u = (21T)
f
p(x,
)ei U(
) d
is absolutely convergent, and it is okay to differentiate under the integral sign. That P(x,i) = C; (r2)
C"'ir2) follows. In order to prove the rest of thE theore m, we need a lemma.
Lemma:
Let
m ... p e Sp,t5(r2), v e Co (r2).
n ThenV
, T1 e:IR
-21-
II v (xl ( E,;) i dxl
lE,;I )m+ON -N p x, e C N (1 + (1 + I T'I I ) Proof of lemma: Integrating by ,a!'ts y:..clis
I T'l a J v(x)
i I J a idX/
< C lal (1 + I E,;1)m + t5lal
since ID
p(X,I;;) I
C(l + 11;;1)m + t5161
To complete the proof of the theorem, we show that the functional
v

,
is well defined if u e t'
(3)

(21T)-n J P(x,l;;) /' (l;;)ei v(x) dl;; dx u =(21T)-n J Pv(l;;)
(I;;) dl;; where Pv(l;;) = f v(x) p (x, 1;;) i dx. For (3) to make sense e for u e ,.' ((2), we only need that Pv(l;;) be rapidly decreasing.
But by the lemma, with I;;
T'I , we have
I p (I;;) I .:: C (1 + I I;; I ) m+ ( t5 -1) N v - N
since t5 < 1, we see that P v (l;;) . is rapidly decreas
ng, as desired, so p(x,D)U eL
((2) is defined for u e [. ((2) .
Definition: If P e s
,t5((2), then the operator p(x,D) defined by (1) is called a pseudo differential operator, and we write
p(x,D) e PS(m,p,t5) .
In the remaining sections of this chapter we examine some of the
fundamental properties of pseudo differential operators. Our treatment follo,om closel
r tI:::Jt of Hormander [34]and [39], where many other
results are proved, and we urge the reader to look at these papers.
-22-
2.
The pseudo local property. , 00 (
K e t f ' (r2xr2), tr-en the map K:C O Ut}--) IV ' (r2)
If
is given by = 0 we have
(1) sing supp p(x,D)u C. sing supp u .
Proof: We show that the distribution kernel of p(x,D) is
smooth off the diagonal in r. x r2. Call
he Kernel K
K(u III v)

1
lJ!v(x) p(x,E,:) ei
(E,:) dE,:dxl 1J!
(-T1) P f (T1 - E,:,J)
(E,:) dE,:d
I < C N J! I
(-y)
(E,:) I (1 + Iy - E,:I)-N dE,:dy
< c' II vii 2 L
II ull 2 L
if N is taken large.
This completes the proof. Note that it breaks down for nonzero t5.
The case of nonzero t5 is somewhat more subtle. The following ingenious proof is due to Hormander [39]. Even though we will
make no use of this more refined continuity result in this course,
we shall give the proof here,
- 38 -
first because it is a very beautiful variation on the idea
that a positive linear functional A on C(K), the set of
continuous function on a compact Hausdorff space K, is
automatically continuous, with II All = A(l), and second because
a good bit of the work we do here would have to be done anyway,
and we shall see the following lemma revealed in section 8
a as Garding's
inequality. a peS z (n), t5 < p, and if p,u
Re p(x,E,:)
> C > 0,
Lemma: If
then there exists a B e PS(O,p,t5) such that
Re p(x,D) - B* B e PS(--,p,t5).
Proof: We write the symbol b(x,E,:) - l bj(x,E,:) as an asymptotic series, with b. e s-j
p-t5) We start with ] p,u
b O (x, E,:) = IR e p (x, E,:) .
a By exercise 2 of this chapter, b O e Sp,t5 (n). Furthermore, the formulas for adjoints and products show that
Re p(x,D) - bO(x,D)* bO(x,D) e PS(-(p-t5),p,t5).
Proceeding by induction, suppose we have the terms
in the asymptotic expansion. We need b j + l e such that
b O "" ,b j S- (j+l) (p-t5) p,t5
(n)
(*)
Re p (x, D)
((b O *
+ . + b. *) + b
1 ) . . ] ]+ + b. 1) + R. 1 ]+ ]+
«(b O +.. .+b j )
- 39 -
with Rj+l e PS(-(j+l) (p-
),p,
). The right hand side of the above expression is equal to
* * * R. + b. 1 (b O +" .+ bj+b j + l ) + (b O +...+b j + l ) b. 1 ] ]+ ]+ * * R. + b. 1 b O + b O b j + l + S. ] ]+ ]
where R. e PS (-j (p-
) , p,
) ] at the previous stage of (*), and S. e PS(-(j-l)(p-
),p,
). ] Then it only remains to set b j + l j b O - l Rj' and the induction is complete. It is easy to verify that the
i
the analogous remainder term
resulting operator B = b(x,D) has the desired property.
Theorem 2: Let A e PS(O,p,
) and assume that
< p.
Suppose
lim sup PA(x,
) I < M < -. I
I+'"
If Keen, there is an
R e PS(-"',p,
) such that, for all u e CO(K), (Au,Au) < M 2 IIul1 2 + (Ru,u)
Proof: C== M 2 - A*A has principal symbol a C == M 2 - laA(x,
) 1 2 , so by the previous lemma there is a B e PS(O,p,
) such that
* 2 * * C - B B = M - A A - B B = RO e PS(-"',p,
).
:JAu,AU)
(Au,Au) + (B
Bu)
= M 2
2 Ilull + (ROU'U)'
and the proof is complete.
- 40 -
Corollary: If A e PS(O,p,
) 2 2 for KC.CI1, A:L (K) .. L l (11) oc
is properly supported, then 2 ( .. L 2 ( cont. Hence A:L loc 11) loc 11)
continuously. Furthermore, if lim aA(x,
) = 0, then I
I"'"
A L 2 (K)" L 2 l (11) is compact. oc
Exercise B: Prove the proposition stated at the end of section
2 in chapter 1. Exercise 9: Define AS on A I QRn) by
s (Asu)"" (
) = (1 + I
r)! u(
).
Show that AS:HtQRn) .. Ht-sQRn) isomorphically. Using these
operators, show that if A e PS(m,p,
) is properly supported
then
s s-m A:H loc .. H loc '
if
< p.
In the proof of L 2 continuity given above, the assumption
< p was necessary in order to ensure that the asymptotic
series defining B was indeed an asymptotic series, i.e. that
the order of the terms went to
Recently, Calderon and Vaillancourt [88] have sho'.,n that A E PS(O, t, t) is continous on L 2 . Beals and Fefferman (86) have made incisive use of this, in tae CBLe I =
. For further
generaliz
tions, see [87j.
Note
ubL the proof of theorem 1
orks for
e
E(J,J,O).
- 41 -
7. Far:tilies of pseudo differential operators. We make S mz(n) into a Frechet space py means of the p,u
seminorms
IpIK,0.,6
:
p
(x,
) (1 + 1
1)-m+plo.l - t51611
where K is a compact subset of n, and 0. and 6 are
multiindices.
We leave the following assertions as exercises for the
reader. We shall make use of them later on, especially in
Chapters V and VI.
1. The map p + p(x,D) is a continuous linear map from
m Sp, t5(n)
to
(H S (n) , comp
2.
Consider
PE(X,
)
H s - m (n) ) if loc ' _E I
1 2 e
t5 < p.
Show that
{p : 0 < E < l} E -
- 0, PE e Sl,O OR ). o n is a bounded subset of Sl, 0 OR ).
In this case, what one must show is that
ID
PEl < Co.(l + 1
1)-10.1
(0 < E < 1)
where the constants C are independent of E. 0. If X is a differentiable manifold which is paracompact,
we give PS(m,p,t5:X) the topology induced from symbols obtained
by local coordinate representations. We leave a precise formulation
to the reader.
- 42 -
A Friedrichs' mollifier on X in a set of pseudo
differential operators
J , 0 < E _ < 1, E -
such that
(i)
J E e PS(-...,l,O)
for E > 0,
J is properly supported. E
(ii)
J u..u in L 2 forueL 2 ,as ("'--'0. E camp {J :0 < E < l} is a bounded subset of PS(O,l,O). E --
(iii)
3. Using partitions of unity subordinate to coordinate coverings
of X, and the result of exercise 2, show that X has a Friedrichs'
mollifier.
4.
Let A e ps(m,p,t5:X), 1 -p
t5 < p.
If
J E
is a Friedrichs'
mollifier on
X, show that [A, J ] E
AJ - J A has the following E E
properties:
(i) [A,J E ] e PS(-"',p,t5) for E > ,0. (ii) [rA,JE]:O
E
l} is a bounded subset of PS (m-(p-t5),p,t5).
5. For convenience, we take X to be compact in this exercise. Let p(x,D) e PS(l,l,O:X), f e L 2 (X). A function u e L 2 (X) is
said to be a weak
solution of the equation
(*)
p(x,D) u
f
if this equation is valid, when p(x,D) is applied to u in
the distribution sense. On the other hand, it is said to be
a strong solution of (*) if there exists a sequence
u. .. u in ]
L 2 (X) with u. e c"'(X), such that p(x,D) u. ] ] Clearly every strong solution of (*) is a weak
f. .. f in L 2 (X) . ] solution.
- 43 -
Making use of the smooth functions u E = JEU, prove that, conversely, every weak solution of (*) is a strong solution. This Friedrichs' mollifier technique was introduced by Friedrichs [23] in order to prove such "weak=strong" results. The technique has also proved a useful tool in passing from
priori estimate to regularity theorems as we will see in Chapter V. For weakcstrong results on manifolds with boundary, see [71] and [67]. Such a result is also implicitly contained in [75].
- 44 -
B. Garding's inequality. This inequality, first proved by Garding [26] for differential
operators and by Calderon and Zygmund for singular integral
operators
oes far beyond being a tool to solve the Dirichlet problem for strongly elliptic operators. Asa tool for proving
energy inequali ties for hyperbolic equations. (a task to which the tool was originally applied by G£rding [27]) and for proving
other important
priori inequalities, it must rank as one of
the fundamental results of the theory.
Theorem: Let A e PS(m,p,t5) and assume 0 < t5 < p < l. - - Suppose that Re °A(x,E,;)
clE,;l m for E,; large, C > O. Then for any real s, for any fixed compact set K, and all u e C; (K) , we have
(1)
2 2 Re(Au,u) > Co Ilull m/2 - C l Ilull s' with Co and C l
independent of u.
Proof: This is a consequence of the lemma proved in section 6, 1 . h A -m/2 A A -m/2 , app 1ed to t e zero-order operator p(x,D) m/2 m/2 since then (Au,u) (p(x,D) A u, Au). There is also a sharp form of Garding's inequality which
says that under the weaker hypothesis that Re 0A(x,E,;) > 0, if A e PS(m,l,O,) it follows that, for K compact,
2 Re(Au,u)
-C l Ilull (m-l)/2
u e C; (K) .
We shall return to this in Chapters VI and VII.
Cha p ter III.
Elliptic and Hypoelliptic Operators
1. Elliptic operators.
Definition: p(x,D) E PS (m, p,o;X) is elliptic of order m if for
each compact K eX, there are constants C K and R such that
I p( x, s ) I > C0 1 + I s I ) m
if
Isl>R, xEK.
Then the Laplace operator
L:.
a 2 2 aX l
+ ." +
a 2 ax 2 n
is elliptic
of order two, since (f L:. (x,S)
_ S 2 1
S2 n
I s I 2
but the
wave operator
o
.0. n-l
i
with symbol
(f 0 (x, S)
_ I s 'I 2 +
2.
is not
" 2 ox n elhptic.
Definition: A parametrix
P
for the operator
Q ( PS(m, p, 0) is a
pseudo-differential operator which is a right and left inverse for Q modulo
smoothing operators:
PQ - I
Kl E PS( -(0)
QP - I
K E PS(-oo) . 2
Exercise 1.
If q(x, D) is elliptic of order -1 -m q(x, S) E S for large S . p,o
m,
q(x, D) E PS(m, p, 0) ,
show that
THEOR EM:
If q(x, D) E PS(m, p, 0) is elliptic,
0< p , then there is
a properly supported
P E PS( -m, p,o) which is a parmetrix of q(x, D).
Proof. We will write the Sf mbol p of P p(x, D) as an asymptotic sur
46-
P - L;p. and get successive approximation
-' to p. First, set J
Po (x,s)
-1 s(x,s) q(x,s) E
-m S p,o
where S vanishes in a nbd of the zeros of q, and is identically for
large S. The formula for a product yields
(f POQ S m-( p -0) S _ 1 E p,o
L;.!. a
(a) Po (x, S) q(a) (x, S)
1 + S (x, S) -1
where
Similarly, we have
(f _ 1 + (PO + Pl)Q
S_l(x,S)+ L; 0'1 a 0
pia) (x, S) q(a) (x, S) + PI (x, S) q(x, S)
Then we take
-1 PI (x, S) =: - S -1 (x, S) q(x, S) , for large S, yielding S m Z( p - 0) p,o
(f(PO+Pl)Q = l+S_Z; S_ZE
In general, if at stage
we have
(f (P O + ... + P. )Q J-l
1 + S . (x, s) -J
S .E S m-j(p-o) - J p, 0
-1 set p.(x, S) =: - S . (x, s) q(x, S) J - J
for large j , and continue. It is
easy to verify that
p
L;p. J
is a right parametrix of Q, i. e. PQ - I =:
Kl E PS( -(0) .
Similarly, one can construct a left parametrix
1 PEP S( - m, p ,0 )
such that
1 QP - I = KZ E PS(-oo) 1 PQP
However, note that
1 (1 + K l)P
pI + KIPI and
1 PQP = P(l +KZ)
P + PKZ .
- 47-
Hence
1 1 P - P = Kl P - PK 2. E PS( -(0), so P itself is a two-sided
parametrix of Q
As a consequence, we have the following regularity theorem.
Corollary.
Let QE PS(m,p,o) be elliptic. If UE f3"(r2) and if
Qui w
00 C (co),1..) an open subset of r2 ,
then u I is smooth. w
Proof:
If P is a properly supported parametrix of Q , we can assume
further that the restriction of the distribution kernel of
P to
wX w
is properly supported.
Then if
00 f = Qui E C (w), w
we have
Pfl = PQ ul u + K ul w w w ul E Coo(w), as asserted. w
Since Pf E Coo(w) and K is smoothing,
The crucial propery of the parametrix used in the proof of this
corollary is that its distribution kernel is smooth off the diagonal.
- 48-
2. Hypoelliptic operators with constant strength.
In the last section, we showed that elliptic operators satisfy a
certain smoothness property. There is a more general class of operators
which sat isfy such a property, of which the heat operator
a at
- .0.
is
another important example, which we now discuss.
Defini tion.
A properly supported pseudo-differential operator
PEP S( m, p , 6; f2) is hypoellipti c if for any open we Q ,
Pul E CCD(w) w
implies
ul E CCD(w), for any u E D' (Q) w
THEOREM 1.
Let P( s) be a polynomiaL The following are equivalent.
(1) P(D) is hypoelliptic.
( 2)
p(a)(s) P(S)
< C(l + Is I)-pial
for 1 s 1 large.
(3) If V = {s E a:;n: P(s) = o} then 1 Re sl
CD, S E V
1 Imsl
CD S E V, 1 s 1 large,
(4) There exists p> 0, C> 0, such that fQr
1 1m S 1 :::: Cis 1 p
Proof.
First we show that the inequality of part (2) implies that P(D) is
hypoelliptic. In fact, we have done most of the work needed to prove this
in Chapter 1. The reader should verify that, if (2) is satisfied, then
-1 0 P(S) E S 0 for large S. The proof that (2) =} (1) is then easily p, cornpleted by arguments we've seen in the last section.
- 49-
Next we show that (1) implies (3). For this, let
--f" = {u E C(r2) : P(D) u = o} C. C(r2)
= {u E C l (r2) : P(D) u = o} C C l (r2)
if P(D)
hypoelliptic
Since
.AI' is a Frechet space under topologies induced from either
C(r2)
1 or C (r2) , and since one is stronger than the other, the open
mapping theorem implies these two topologies coincide. i :: 0, we immediately
see that Q satisfies condition (2) of Theorem 1; hence, Q(D) is hypo-
elliptic.
It remains to prove the inequality (2). We shall obtain this as the second
of the following two lemmas.
There are constants C l' C 2 such that, for all polynomials R of degree .:::.K, we have, with S E m n , S E m n , t > 0 ,
Lemma 2:
CIR(S,t) < sup IR(s+s)1 < C 2 R(s,t) Isl < t
Proof:
The right-hand inequaltiy follows from the Taylor expansion
of R( s + s) about S. Since, if we set Rt( S) = R(tS), we get R( S, t) =
-- Rt(s/t,l), it suffices to prove the other inequaltiy with t = 1 Pick N distinct points sl'" . ,sN in the unit ball in m n , where N is the dimension of the space of polynomials in a:; [X l' . . . ,X n of
degree .:::. K. Then the equations for the coefficient of polynomials
P.. J
gi ven by
P / sk) = 6 jk
- 53 -
can be solved uniquely for polynomials of degree < K, and we obtain
N R(T]) = L; j=l
R(I;,.) P .('11) J J
known as the Lagrange interpolation formula.
N R(S + '11) = L; R(S + 1;,.) P.(T]) j=l J J R(a)W N p(
)(O) R(S + 1;,.) = L; j=l J J
The left-hand inequality. with t = 1 . now follows easily.
Lemma 3:
'" Inequality (2) above is valid: Q(s. t) < C P(S. t) for t> 1 , all S
Proof:
Using Lemma 2. and the boundedness of
Q(s) P(S)
for large S,
Q(s.t)
<
C l
sup II;,I < t
I Q(S + 1;,) I
< C 2 (1 + s up I P (S + 1;,) I ) II;,I < t
<
C 2
sup P( s + 1;,. 1) II;, I .:: t
< C 3 sup P(S+I;,) < C 4 P(s.t+l) II;,I < t+l
< C 4 (1 +.;.)m P(S,t)
Thus the proof of proposition 1 is complete. In particular. it follows
that if p(x. D) is formally hypoelliptic of constant strength, then any
- 54 -
constant coefficient operator P O(D) = p(x O ' D) obtained by freezing
the coefficients of p(x, D) at some point, is hypoelliptic.
The following corollary of proposition 1 will be of interest.
Lemma 4:
Ifl P(D) is hypoelliptic and
Q(s) P(S)
is bounded for large S,
then
Q(a)w P(S)
< ci sl-p1a l
for large
s .
Proof:
Pick £
so small that CIP(S)':::' P(S) + (Q(S)':::' CZP(S)
for large S. By proposition 1, Pl(D) = P(D) + [Q(D) is hypoelliptic.
Hence we have
Q(a)w p(a) (S) - p(a)W 1 = P(S) £ P(S) P (a)( S) P 1 (S) p(a)W 1 = - P 1 (S)
P(S) £ £
< ci sl-p1a l
for S large.
Suppose now that p(x, D) is a formally hypoelliptic operator with
constant strength. Freeze the coefficients at one point
Xo
to obtain a
hypoelliptic constant coefficient operator P(D) = p(x O ' D). Let E be a
properly supported pseudo-differential operator, E E PS(O, p, 0), such that -1 (f E(x, S) = P( S) for large S. A s we have seen in the proof of Theorem 1,
- 55 -
E is a parametrix of P(D) .
Proposition 2:
p(x, D) E and E p(x, D) belong to PS(O, p, 0) and are
elliptic.
Proof:
sO p, 0 estimation, using Lemma 4. From this we have p(x, D)E E PS(O, p, 0)
That
-1 p(x, S) P( S) E
for large S
is a routine
That E p(x, D) E PS(O, p, 0) follows from looking at the terms in the
asymptotic expansion of its symbol, and we leave the details to the reader.
That the operators are elliptic is obvious.
Now for the second main result of this section.
THEOREM 2:
If p(x, D) is a formally hypoelliptic operator of constant
strength, then p(x, D) is hypoelliptic.
Proof:
If A E PS(O, p, 0) is a parametrix for the elliptic operator
E p(x, D) then (AE) p(x, D) - IE PS(-oo, p, 0)
so AE is a left parametrix
of P (x, D) , and the argument of the previous section shows that p(x, D) has
the required regularity property.
- 56 -
1.
Let A(x, D) =
L; I al < 2
a (x)D a a
be a second order operator, with
I a Dj = i ax. ' which is strongly elliptic in the sense that J
Re L; lal =2
a (x) sa> a
ci s I 2 .
We assume the coefficients are smooth; they could also depend on t
P rove that
a at
- A(x, D) is formally hypoelliptic with constant strength.
2.
Consider the hypoelliptic operator
_ a a 2 JR 2 P(D) - Y - 2 on U ax change of coordinates of JR 2
Prove that there exists a non-linear
with respect to which P becomes an operator which is not formally
hypo elliptic , Is this new operator hypo elliptic ?
3, Let X and Y be two smooth vector fields in JR 2 which are
linearly independent at each point. Prove that the second order operator
X2 _ Y is hypoelliptic.
4. Let p(x, D) be a formally hypoelliptic operator with constant strength.
Prove that its adjoint
p(x,D)
also enjoys this property.
In general, the adjoint of a hypoelliptic operator has the local
solvability property. See Treves [77] ; a similar argument, in another
context, is given in Chapt er VI of these notes.
- 57 -
33. References to further work.
There is a more general notion of operators of constant strength than that presented here; see [ 31]. It is known ([76] ) that every hypo- elliptic operator of constant strength is formally hypoelliptic. However, there are other hypoelliptic operators which are not of this sort. Treves and Ho rmander have examined a class of operators that one may call formally hypoelliptic with slowly varying strength. In fact, the classes PS(m, p, 6) with 6 > 0 , which we have not used, were invented to handle these operators; see [34] .
There is a nearly complete characterization of second order hypo- elliptic operators with real coefficients. See [36] and, for a simpler
treatment using pseudo-differential operators, see [43] treated in [63] . There is also a theory of hypoelliptic operators of principal type, for
This topic is also
which we refer the reader to [79] .
hapter IV. The Initial Value Problem. Hyperbolic Operators
In this chapter our main aim is to stu8y the Cauchy problem for
hyperbolic operators. That is, if L(x, t, D) is a hyperbolic operator
of order m, we wish to solve the initial value problem
Lu = f u(x,O) = gl a u(x,O) at = gz a m - l u(x,O) = gm m-l at
We assume
L =
am at m
m-l L; j=O
a j A . (x,t,D) . m-J x at J
where A . m-J
is a differential operator in the x variables of order m-j, depending
smoothly on the parameter t. The symbol of L is
L (x,t, S, T) = (iT)m -
m-l L; j=O
A . (x, t, s) (iT)J m-J
There are several notions of what it means for such an operator to
be hyperbolic. We shall introduce a couple of them in later sections, as
we attempt to solve the above initial value problem.
- 59 -
l. Reduction to a first order system,
Suppose the coefficients of L a re smooth and defined on all of m n .
In that case, with
(1\u) II (S) ::: (1 + I sl 2) 112
(I;), write
u l
m-l 1\ u
U z =
') m-2 1\ u <1*
... j-l m-j u. ::: (-) 1\ u J 01* :) m-l u m-l u m ;)t Then the equation Lu :: f turns into the fi r s t order system
u l 0 1\ O. 0 u l 0 u l 0 0 0 1\. 0 0 0 a + + at :: K 1\ 0 0 u b l b 2 b 3 . .b u f , f m m m \
where
b :: j
j-m A . (x, t, D ) A ,Now m-J + 1 x
K E PS( 1, 1, 0)
and it has a
principal symbol, which we call (fK (x, t, s) which is a matrix valued o
function, homogeneous of degree 1 in I; Furthermore, as the reader can
verify, the eigenvalues of (fK (x, t, s) differ from the roots T 1 (s), . . . , T m (I;) o of L (x, t, S, T) ::: 0 by a factor of i, where L is the principal part of m m
L, consisting of all the terms of L of order precisely m.
- 60 -
of
u
is equivalent to specifying u l ...., u m
oj o t j at t = 0 .
u(x,O),
j=O,....m-l
It is clear that specifying the Cauchy data
Now, to see when we can solve the initial value problem
o a-t u =
Ku + f
u(x,O)
g
with K E PS(l. 1,0) a matrix-valued operator, let us take a look at the
constant coefficient case, (fK (x, t, 1;) = k(I;). With f = 0 ,
(s. t) =
- n f i(xl;} (21T) u(x, t) e ' dx,
we have
G. (I;, t)
et k(l;) g(l;)
1£ we require that the initial datum
2 n 2 n gEL (JR ) leads to u(', t) E L (JR )
for each t E JR, we need
t k( 1;) e
bounded in 1;. for each real t. 1£
k(l;) is homogeneous of degree 1 in 1;. this is equivalent to the assertion
that. for each I; I O. k(l;) is diagonalizable. and all its eigenvalues are
pure imaginary.
1£ we recall how the roots of L (x, t. S. T) = 0 are connected to the m
eigenvalues of
(f ,. (x. t. 1;) . the following definition becomes reasonable. hO
Definition:
L
is hyperbolic if the roots
T (x. t. 1;). . . . . T (x, t, T) 1 m
of
L (x, t. 1;. T) = 0 are all real. m
Hyperbolicity in the broad sense will not allow us to solve the Cauchy
problem. but in the next two sections we introduce further conditions which
will.
- 61 -
In the next two sections, we shall suppose, for technical convenience,
that the coefficients of all operators are periodic with respect to the x variables, so that our unknown solution u is to be defined, say. in [O,T]Xr n , where Tn isthen-torus. The main point of sections 2 and 3 is just to get local existence theorems. We get good global existence theorems by the finite propagation speed/ finite domain of dependence argument of section 4.
- 62 -
32. Symmetric hyperbolic systems.
Definition: The operator
a at - K is symmetric hyperbolic if
K + K'" E PS(O, 1,0), where K = K(t) is a smooth one-parameter family
of operators in PS(l,l,O).
We shall prove that a solution of the initial value problem exists and
is unique by proving an a priori inequaJity for such solutions. Existence will
then follow, by a little functional analysis. The following inequaJity from the
the theory of ordinary differential equations will be a useful tool.
Lemma (Gronwall's inequality): If y E e l
and
, Y (t) + f(t)y
g(t) , then
(1 )
y(t) <
t - fo f(T) dT e
[ YO + Jot
g(T)
T JO e
f(X,) d X,
d TJ
P roof: The hypothesis is equivalent to the inequality
d dt
(y e Jot f)
Jot f < g(t) e
and integrating this yields (1) .
co ( n+l ) Suppose now that u E eO '[' differentiate II u(t) II 2 = I I u(x, t) I 2 dx ,
and
a u _ Ku = f . at
If we
we respect to
t , we get
d (u' ,u) (u,u') dt (u,u) + = (Ku + f, u) + (u, Ku + f) = ( (K + K
') u,u) + 2 Re (f, u) ell ull 2 2 -', < + ell fll , since K +K E PS(O, 1,0) .
- 63 -
Applying Gronwall's inequality to this yields
T (2) II u(t) II 2 .:: e'll u(O) II 2 + e' Jo II f(T) II 2 dT, 0 < t .:: T,
where e = e (T) is independent of u, f, and t, for 0 < t < T.
Exercise 1: Differentiating
S 2 II A u(t) II ' prove that, for all real S ,
(3)
II u(t) II
.:: ell u(O) II
+ e J
II f(T) II
dT, 0 < t < T,
where
II II S is the norm in HS('['n), LAS,
E PS(S, 1,0) .
given
co n UE eo(IRXT ) ) u
- Ku = f .
Hint:
1 , S+l n More generally, (2) and (3) are valid for u E e ([ 0, TJ, H ('[' ) ) , S n S+ 1 n fe e([O,T], H ('[' )), 0 , £
K£ = KJ£ in a continuous operator in H S (1['n). Then (,:,
,) can be
- 64-
regarded as a Banach-space valued ordinary differential equation, to which
the Picard iteration method applies (see [15]). Then, given
S n 1, are pure imaginary and distinct, the operator
a at- K
is called strictly hyperbolic.
In analogy, we have the following.
Definition:
If
L is a differential operator of order mt L is said to be
strictly hyperbolic with respect to the initial surfaces t = const. if, for
each fixed (x, t, S), the root s
T I'
, T of L (x, t, S, T) m m
o are real
and distinct, provided S # 0
The way we treat the Cauchy problem
a at U =
Ku +f
u(O)
1 .
( ii)
(f = RK
(f RK
". mod S
0 ('['n) . ,
- 67 -
Proposition: Any strictly hyperbolic first order system
-L - K at
has a symmetrizer.
Proof:
, If we let the eigenvalues of (fK (x, t, S), o ).. 2(x, t, S) < ...
for
I s I > 1 ,
be
i).. (x, t, S), v
where
).. 1 (x, t, S) <
<
)..k(x, t, S),
K being
a k-by-k system, then).. are well defined CCO functions of (x, t, S), v
homogeneous of degree 1 in S.
Hence ).. v
1 ( 'I'n ) Sl,O
for large S.
Similarly, if
P (x, t, S) are the projections onto the associated v
eigenvalues of
i)..)x,t,S), IV
1 21Ti
-1 I (q>- (fK (x,t,S)) d1;, then Yv 0 k "_ R(x, t, D) L; P .(x, t, D) 0' P .(x, t, D) j= 1 J J
P v
o n E Sl, 0 ('[' )
for large S .
Now let
It is clear that R is a symmetrizer, and R E PS(O, 1,0). Note that
(f RK(x, t, S)
k = i L; j= 1
).. .(x, t, S) J
P.(x,t,s)
'P.(x,t,s) mod S
0 J J ,
With such a symmetrizer, we can prove an energy inequality. In fact,
let R 1 = R mod PS( -1, 1,0), R 1 .:::. 11 > 0 , positive self adjoint operator on L 2(T n ). That we can do this follows from G.frding's inequality. Hence
d dt (R 1 u, u)
I ( R 1 u, u) + (R 1 u " u) + (R 1 u, u ')
I = (R 1 Ku + f, u) + (R 1 u, Ku +f) + (R lu, u)
=
I (R 1 K+K"'R 1 )u,u) + (f,u)+(R 1 u,f) + (R 1 u,u)
.s. c II u II 2 + C II f II 2
since property (ii) of the symmetrizer yields R 1 K+ K'cR 1 E PS(O, 1,0).
- 68 -
Gronwall's inequality applied to this yields again our energy inequality 2 2 t 2 II u(t) II .s. C II u(O) II + C J II f(T) II dT, 0 < t < T, and similarly, o
for all real S,
2 II u(t) II S .s.
C s II u(O) II
+ C s Jt II f(T) II
o
dT,
o < t <
T,
valid for any
1 S+l n UE C ([O,T]. H ('['))
, if u = Ku + f .
From this energy inequality, one derives, in exactly the same fashion
as in the previous section, the following existence and uniqueness theorem.
THEOR EM 1
If
a at
- K is strictly hyperbolic, then the Cauchy problem
(1 )
a - u = Ku + f at
u( 0) H I s I
for all (T, S)
with
Lm(x O ' to' S, T)
o ,
s = 0 .
A
I I v I
smooth surfact S
n+l. . in JR is said to be space like for
L
at (x O ' to)
if each normal vector to S is timelike.
Proposition: If S is space like for the strictly hyperbolic operator L,
and if v is the normal to S , then the equation
L(x O ' to' W+ TV
o
has
m
di stinct real root s
T l ,"', Tm '
if
W
is not proportional to v .
Also, S is noncharacteristic for L.
The proof is fairly straightforward and will be left to the reader.
- 71 -
From this we have the following important fact. If S is a space like
surface, one can construct a smooth real valued function
I t (x, t)
such that
S is a level surface for
I I t , and such that every level surface of t
is
space like for L. Furthermore, if L is expressed in terms of the new
coordinate s
I (x, t ),
L is strictly hyperbolic with respect to the hypersurfaces
I t = con st. Thus the Cauchy problem Lu = f can be solved, locally, with
Cauchy data given on S.
- 72 -
4. Finite propagation speed; finite domain of dependence.
Let L '" L(x, t, D , D) be a differential operator of order m x t n+l on JR , with smooth coefficients, which is strictly hyperbolic with
respect to the hyperplanes t'" const. Suppose
Gis a bounded open
subset of {(x, t) € JRn+l: t > O} whose boundary oG consists of two
parts, Sl '" 0 Gn JRn and S2' a space like surface. The principal
result of this section is the following uniqueness theorem.
S n THEOREM 1: Let u € C(JR, H (JR ) ) , and suppose that u(O) '" 0 on Sl
while Lu '" 0 on G. Then u
vanishes on G.
Proof:
What we will show is that {u,

be a local solution to the Cauchy problem
L"" '"

1 ... =

I'" 0 m-l S2 OV
-73 -
"" We can continue to be zero above 8 2 , so is a smooth function satisfying L -:I\ = 'P, vanishing in a nbd of the closure of 8 2 , Thus we can
write
o = < , Lu)
*" = , u)
or u, together with all
Cauchy data, vanish on or2, vanishing in a nbd of the corner of this
boundary.
This type of argument was first used by Holmgren to prove uniqueness
in the Cauchy problem for differential operators with analytic coefficients, and
non-characteristic boundary. 8ee[13] or [31] ; in that case, the existence
theorem used was the Cauchy-Kowalevski theorem.
Definition:
L ( ) E m n+l et so,t o
If there is a bounded domain G with
(x o ' to) E G , whose boundary consists of a portion 8 1 of m n ::; [(x, t) :t=O }
and a space-like surface
8 2
for
L, we say
8 1
is a domain of dependence
for (x O ' to)' and we say that (x O ' to) has a finite domain of dependence. The following theorem is a simple consequence of Theorem 1 together
with the results of the previous section.
THEOR EM 2:
Let L be a differential operator of order m, strictly
hyperbolic with respect to the surface t = const.
(a) If the vector (v', v 0) n+l (x O ' to) E m
is timelike whenever
IVol
> M, then
any
I v' I has the bounded domain of dependence
BMlt o I (x O ) =
{XE m n : I x-xol < MltoD
-74 -
(b) Given 'P. E
I(JRn) , f E C(JR,S '(JRn) ) , the Cauchy problem J
Lu f
cp( 0) :::: 'PO
m-l a m-l CP(O) at
'Pm_l
has a unique solution u E C(JR, ':tJ '(JRn) ) ,
n+l if each point (x O ' to) of JR
has a finite domain of dependence.
(c) If the conditions of (a) are satisfied, and if supp 'P. C K, f:::: 0 , J
then the solution of the above Cauchy problem has the property that, for
each t E JR , supp u(', t)
KMltl '
where
K = {XE JRn: dist (x,K) < r}. r -
The last property is known as finite speed of propagation of a signal.
Exercise 5: Suppose
L::::
am at m
m-l + L; j::::O
a j A . (x, t, D ) :;- m-J x ut. J
is a differential
operator of order m, strictly hyperbolic with respect to the surfaces t:::: const.,
n+l and suppose every (x O ' to) E JR has a finite domain of dependence. Then n/2 - £ + m -1 n there is a unique R E C(JR, H (JR )) such that loc
LR 0
R(O) 0
m-2 a m-2 R(O) :::: 0 at m-l L- R(O) :::: 0 m-l at
-75 -
Let E £; I(JRn+l) be defined by (t) = R(t) for t> 0, (t) = 0
for t < o. Prove that L = 6. What can you say about the support
of ?
Exercise 6: Concoct an example of a strictly hyperbolic operator L
whose coefficients near infinity behave so badly that not all points n+l (x o ' to) E JR have a finite domain of dependence.
76 -
5. The vibrating membrane problem.
In this section we shall merely indicate how to obtain a hyperbolic
equation for one physical process. For a further introduction to the
equations governing continua, see Goldstein [2 ] .
We suppose we have a thin membrane, pulled tight, which vibrates
slightly in the direction perpendicular to the plane in which it is
to lie; call it the x-y plane. If cp(x, y) is the distance above or below that
plane, then for q> small, Hook's law says the potential energy in the
membrane, due to the slight displacement it has undergone, is approximately
2 V = I kl vq>1 dV r2
where r2 is the plane occupied by the membrane and k is an appropriate
factor, k> 0 on r2. On the other hand, the kinetic energy due to this
vibration is
2 T = I plq>t l dV r2
where p is half the mass density. Now Hamilton's principle says that j t l the action to
(T- V) dt is stationary. That is, we are led to the calculus
of variations problem
d d I(q> + 11lJ;)1 _ 0 = 0 11 11- 00 rl ljJ E Co (r2 X (to' t l)) where I( q» = J to
f f k 1Vq>1 2 r2
pi q>t12 J dVdt.
for all
-77 -
Hence we get, for all
co lj; E C 0 (iJ X ill) ,
d o = d
I(